Question

Find each sum or difference. Give your answers in lowest terms. If an answer is greater than \(1,\) write it as a mixed number. $$\frac{3}{8}+\frac{2}{3}$$

Short answer

1 \(\frac{1}{24}\)

Step-by-step solution

Find a Common Denominator

The denominators of the fractions are 8 and 3. The least common denominator (LCD) of 8 and 3 is 24. Determine the LCD by finding the least common multiple of the two denominators.

Convert Fractions to Equivalent Fractions

Convert each fraction to an equivalent fraction with the denominator of 24. For \(\frac{3}{8}\): \(\frac{3}{8} \times \frac{3}{3} = \frac{9}{24}\) For \(\frac{2}{3}\): \(\frac{2}{3} \times \frac{8}{8} = \frac{16}{24}\)

Add the Fractions

Now that the fractions have the same denominator, add the numerators: \(\frac{9}{24} + \frac{16}{24} = \frac{9 + 16}{24} = \frac{25}{24}\)

Simplify the Fraction

Since \(\frac{25}{24}\) is an improper fraction, convert it to a mixed number. Divide the numerator by the denominator: \(\frac{25}{24} = 1 \frac{1}{24}\)

Key concepts

least common denominator

When adding fractions, the first step is to ensure the denominators (bottom numbers) are the same. This is called finding the Least Common Denominator (LCD).
The LCD is the smallest number that both denominators can divide into evenly.
For the fractions \(\frac{3}{8}\) and \(\frac{2}{3}\), the denominators are 8 and 3.
The LCD of 8 and 3 is 24 because 24 is the smallest number divisible by both 8 and 3.
To find the LCD, list the multiples of the larger denominator (8): 8, 16, 24... then see which one is a multiple of 3. You'll get 24 as the least common multiple.
Once you have found the LCD, you can convert both fractions to equivalent fractions with this common denominator.

improper fractions

An improper fraction is a fraction where the numerator (top number) is larger than the denominator (bottom number).
After adding the numerators of the equivalent fractions with the LCD, we get \(\frac{25}{24}\).
This result is an improper fraction because 25 (numerator) is greater than 24 (denominator).
Improper fractions are useful, but they are often converted to mixed numbers for simplicity and standard form.
To convert an improper fraction to a mixed number, you need to divide the numerator by the denominator.
Here, 25 divided by 24 gives a quotient of 1 and a remainder of 1, so \(\frac{25}{24}\) becomes 1 \(\frac{1}{24}\).

mixed numbers

A mixed number combines a whole number and a fraction.
Mixed numbers are easier to understand and use in everyday scenarios compared to improper fractions.
After converting the improper fraction \(\frac{25}{24}\) from the given problem, we have identified the mixed number as 1 \(\frac{1}{24}\).
This means that the fraction has a whole number part (1) and a fractional part (\(\frac{1}{24}\)).
Mixed numbers can also be helpful when comparing sizes and performing additional operations, such as addition and subtraction, in a more intuitive manner.
When dealing with sums and differences of fractions that are greater than 1, always remember to convert them into mixed numbers for clarity.